Let $G$ be a group with a finite presentation. The Cayley graph of the finite presentation is a $1$-dimensional cell complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (connecting two elements). Its Cayley complex is a $2$-dimensional generalization, where the $2$-cells are given by the relations (here is a nice YouTube video on it by Daniel Tubbenhauer).

Question: What about a generalization of the Cayley complex to higher dimensions, but still given by the finite presentation (in particular, the $1$-skeleton is its Cayley graph, and the $2$-skeleton is its Cayley complex)?

I am looking for obstructions preventing such a generalization to all the finitely presented groups, and/or references discussing such a generalization to all or a large class of them.

Here is my effort to see what such a generalization might look like.

Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (connecting two elements). The Cayley complex of the finite presentation is a $2$-complex where the $1$-skeleton is the Cayley graph, and the $2$-cells are given by the relations (here is a nice YouTube video on it by Daniel Tubbenhauer).

Let us call Cayley $n$-complex of a finite presentation, the corresponding Cayley graph if $n=1$, and the corresponding Cayley complex if $n=2$, as mentioned above.

Question: What about a Cayley $n$-complex made from a finite presentation for $n>2$ (whose $r$-skeleton is a Cayley $r$-complex, for $r<n$)?

I am looking for obstructions preventing such a generalization to all the torsion-free finitely presented groups, and/or references discussing such a generalization to all or a large class of them.

Here is my effort to see what such a generalization might look like.

A Cayley graph is a $1$-dimensional cell complex associated to any presentation of a group $G$, where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (connecting two elements). A Cayley complex is a $2$-dimensional generalization, where the $2$-cells are given by the relations (here is a nice YouTube video on it by Daniel Tubbenhauer).

Question: What about a generalization of the Cayley complex to higher dimensions?

I am looking for obstructions preventing such a generalization to all the finitely generated (or finitely presented) groups, and/or references discussing such a generalization to all or a large class of them.

Here is my effort to see what such a generalization might look like.

Let $G$ be a group with a finite presentation. The Cayley graph of the finite presentation is a $1$-dimensional cell complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (connecting two elements). Its Cayley complex is a $2$-dimensional generalization, where the $2$-cells are given by the relations (here is a nice YouTube video on it by Daniel Tubbenhauer).

Question: What about a generalization of the Cayley complex to higher dimensions, but still given by the finite presentation (in particular, the $1$-skeleton is its Cayley graph, and the $2$-skeleton is its Cayley complex)?

I am looking for obstructions preventing such a generalization to all the finitely presented groups, and/or references discussing such a generalization to all or a large class of them.

Here is my effort to see what such a generalization might look like.